AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide the student with an elementary introduction to the concepts and methods of Algebraic Topology.

AIMS AND LEARNING OUTCOMES

The aim of this teaching is to provide students with a consolidation of the techniques of algebraic topology already learned in the teaching of Geometry 2, in particular through the theory of universal covering. In addition, the teaching aims to introduce new concepts such as the theory of CW-complexes to the theory of homology and cohomology, in particular by using singular homology and higher homotopy groups as examples, All this without neglecting the motivations and historical background on the emergence of these objects. At the end of the course, the student will be able to calculate coverings of a topological space and quotients of a topological space under the action of a group. Establish whether a topological space is a CW-complex and establish a cell subdivision. Finally, the student will be able to calculate the homology and cohomology groups of simple topological spaces and calculate their cohomology ring.

PREREQUISITES

The teaching is a natural continuation of the teaching of Geometry 2. It is advisable to have taken at least one course in: linear algebra and analytic geometry, general algebra, general topology and an introduction to algebraic topology.

TEACHING METHODS

Lecture

SYLLABUS/CONTENT

1.  Topological varieties and differentiable varieties.
2.  Covering of topological spaces theory and exaples and actions of properly discontinuous groups.
3. Universal coverings.
4. CW - Complexes and their properties.
5. Singular Homology and Cohomology. In particular, the Mayer-Vietoris Theorem and the Universal Coefficient Theorem are presented. 
6. Cohomology rings
7. Higher homotopy groups.

RECOMMENDED READING/BIBLIOGRAPHY

1. M Manetti: Topologia , Springer.

2. C Kosniowski: Introduzione alla topologia algebrica , Zanichelli.

3. W.S. Messey: A basic Course in Algebraic Topology , Springer.

4. Allen Hatcher Algebraic Topology, on-line notes

5. Weibel   Homological algebra, Cambridge University Press

TEACHERS AND EXAM BOARD

Exam Board

MATTEO PENEGINI (President)

FABIO TANTURRI

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral Exam.

Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.

ASSESSMENT METHODS

During the oral examination, the student must be able to: prove the theorems presented in the lecture, correctly state all definitions and solve simple exercises consisting in the calculation of coverings, homology and cohomology.

Exam schedule